Séminaire d'analyse

In this talk, we discuss the effects of perturbations on the topology and geometry of nodal sets/zero sets of Laplace eigenfunctions. A conjecture by Payne states that the nodal set of the second Dirichlet eigenfunction on a bounded planar domain intersects the boundary at exactly two points. We will look into certain stability properties of the nodal sets and discuss some recent results concerning the conjecture. Then, utilising the stability properties, we will observe the prescription of nodal data on Riemannian surfaces, focusing on the following two aspects: the construction of eigenfunctions with a prescribed number of nodal intersections at the boundary, and the realisation of Courant-sharp eigenfunctions at arbitrarily high levels.